Modeling with curves

Transcription

1 Modelin with curves

2 Motivation Need representations of sooth real world objects Art / drawins usin C need sooth curves Aniation: caera paths (C) Dou Bowan, Virinia Tech,

3 Naïve approach Siply use lines & polyons to approxiate curves & surfaces curve: piecewise linear function lots of storae if accuracy desired hard to interactively anipulate the shape Instead, we ll use hiher-order functions Note: difference between stored odel and rendered shape (C) Dou Bowan, Virinia Tech,

4 Approaches Explicit Functions: y = f(x) [e.. y = 2x 2 ]. Only one value of y for each x 2. Difficult to represent a slope of infinity Iplicit Equations: f(x,y) = 0 [e.. x 2 + y 2 - r 2 = 0]. Need constraints to odel just one part of a curve 2. Joinin curves toether soothly is difficult Paraetric Equations: x = f(t), y = f(t) [e.. x = t 2 +3, y = 3t 2 +2t+]. Slopes represented as paraetric tanent vectors 2. Easy to join curve seents soothly (C) Dou Bowan, Virinia Tech,

5 Paraetric Curves Linear: x = a x t + b x 0 t y = a y t + b y Quadratic: z = a z t + b z x = a x t 2 + b x t + c x y = a y t 2 + b y t + c y Cubic: z = a z t 2 + b z t + c z x = a x t 3 + b x t 2 + c x t + d x y = a y t 3 + b y t 2 + c y t + d y z = a z t 3 + b z t 2 + c z t + d z (C) Dou Bowan, Virinia Tech,

6 Joinin Curve Seents Toether 0 eoetric continuity: Two curve seents join toether eoetric continuity: The directions of the two seents tanent vectors are equal at the join point C continuity: Tanent vectors of the two seents are equal in anitude and direction (C unless tanent vector = [0,0,0]) C n continuity: Direction and anitude throuh the n th derivative are equal at the join point (C) Dou Bowan, Virinia Tech,

7 Joinin Exaples R 2 R 0 R TV 3 TV 2 (C) Dou Bowan, Virinia Tech,

8 (C) Dou Bowan, Virinia Tech, Notation x = a x t 3 + b x t 2 + c x t + d x y = a y t 3 + b y t 2 + c y t + d y Q(t) = T C z = a z t 3 + b z t 2 + c z t + d z T = t 3 t 2 t [ ] " % & = z y x z y x z y x z y x d d d c c c b b b a a a C

9 Alternate Notation Q(t) = T M [ t 3 t 2 t ] T Matrix Basis Matrix eoetry Matrix & % Idea: Different curves can be specified by chanin the eoetric inforation in the eoetry atrix. The basis atrix contains inforation about the eneral faily of curve that will be produced. (C) Dou Bowan, Virinia Tech, " & % "

10 (C) Dou Bowan, Virinia Tech, Exaple: Paraetric Line x(t) = a x t + b x x'(t) = a x y(t) = a y t + b y y'(t) = a y z(t) = a z t + b z z'(t) = a z [ ] " % & = z y x z y x b b b a a a t t Q ) ( [ ] " % & = ' z y x z y x b b b a a a t Q 0 ) ( [ ] " % & " % & = z y x z y x t Q t ) ( [ ] " % & " % & = ' z y x z y x t Q ) (

11 Exaple (cont.) Since two points, P & P 2 define a straiht line, then the eoetry atrix should be: = P 2 = P 2. What is the basis atrix? We know that at t = 0, Q(0) = P t = t > Q'(0) = P 2 - P at t =, Q() = P 2 t = 0 P2 Q'() = P 2 - P t < 0 P (C) Dou Bowan, Virinia Tech, 2008

12 Exaple (cont.) Solve these 4 siultaneous equations to find the basis atrix M line : Q(0) = [0, ] M line = P Q'(0) = [, 0] M line = P 2 - P Q() = [, ] M line = P 2 Q'() = [, 0] M line = P 2 - P (C) Dou Bowan, Virinia Tech,

13 Exaple (cont.) 2 x + 22 x 2 = x ( + 2 )x + ( )x 2 = x 2 2 y + 22 y 2 = y ( + 2 )y + ( )y 2 = y 2 2 z + 22 z 2 = z ( + 2 )z + ( )z 2 = z 2 x + 2 x 2 = x 2 - x = - 2 = y + 2 y 2 = y 2 - y 2 = 22 = 0 z + 2 z 2 = z 2 - z &' M = line % 0" (C) Dou Bowan, Virinia Tech,

14 Blendin functions Multiplyin the T and M atrices ives you a set of functions in t; these are called blendin functions. There is one blendin function for each of the pieces of eoetric inforation in the atrix. The value of a blendin function at a certain value of t deterines the effect of the correspondin piece of eoetric inforation at that point alon the curve. ' % & 0 " Line blendin functions: [ t ] = [ t t] f(t) P fn. P 2 fn. 0 (C) Dou Bowan, Virinia Tech, t

15 Herite Curves Defined by two endpoints and tanents at the endpoints. R P4 R4 P (C) Dou Bowan, Virinia Tech,

16 Herite Curve Exaples (C) Dou Bowan, Virinia Tech,

17 Herite Curve Exaples (cont.) (C) Dou Bowan, Virinia Tech,

18 (C) Dou Bowan, Virinia Tech, Herite Matrices Q(t) = T M H H " % & = 4 4 R R P P H T = t 3 t 2 t [ ] " % & ' ' ' ' = M H

19 Herite Blendin Functions Q(t) = T M H H Q(t) = (2t 3 3t 2 +)P + (2t 3 + 3t 2 )P + 4 (t 3 2t 2 + t)r + (t 3 t 2 )R 4 (C) Dou Bowan, Virinia Tech,

20 Bézier Curves (C) Dou Bowan, Virinia Tech,

21 (C) Dou Bowan, Virinia Tech, Bézier Matrices Q(t) = T M B B " % & = P P P P B T = t 3 t 2 t [ ] " % & ' ' ' ' = M B P & P 4 : endpoints R = 3(P 2 -P ), R 4 = 3(P 4 -P 3 ) constant velocity curves if control points are equally spaced

22 Bézier Blendin Functions Q(t) = (t 3 + 3t 2 3t + )P + (3t 3 6t 2 + 3t)P 2 + (3t 3 + 3t 2 )P 3 + t 3 P 4 (C) Dou Bowan, Virinia Tech,

23 eneral Bézier Curves Let P throuh P n+ be points that define the curve. Then: where n " i =0 Q B (t) = P i + B i,n (t), t [0,] B i,n (t) = C(n,i) t i (-t) n-i {Blendin functions} and C(n,i) = n/(i(n-i)) {Binoial coefficient} (C) Dou Bowan, Virinia Tech,

24 eneral Bézier Curves (cont.) For two points, n = : Q B (t) = (-t)p + tp 2 For three points, n = 2: Q B (t) = (-t) 2 P + 2t(-t)P 2 + t 2 P 3 For four points, n = 3: Q B (t) = (-t) 3 P + 3t(-t) 2 P 2 + 3t 2 (-t)p 3 + t 3 P 4 (C) Dou Bowan, Virinia Tech,

25 Bézier Curve Characteristics. The functions interpolate the first and last vertex points. 2. The tanent at P 0 is proportional to P - P 0, and the tanent at P n to P n - P n-. 3. The blendin functions are syetric with respect to t and (-t). This eans we can reverse the sequence of vertex points definin the curve without chanin the shape of the curve. 4. The curve, Q(t), lies within the convex hull defined by the control points. 5. If the first and last vertices coincide, then we produce a closed curve. 6. If coplicated curves are to be enerated, they can be fored by piecin toether several Bézier sections. 7. By specifyin ultiple coincident points at a vertex, we pull the curve in closer and closer to that vertex. (C) Dou Bowan, Virinia Tech,

26 Renderin curves Iterative ethod based on paraetric forula evaluation t=0 plot x(t), y(t), z(t) increent t by a sall at. and repeat until t This is slow, dependin on step size Can increase perforance usin difference ethods (as with line drawin) (C) Dou Bowan, Virinia Tech,

27 Modelin with surfaces

28 Modelin surfaces Extension of paraetric cubic curves called paraetric bicubic surfaces Idea: infinite of curves stacked toether equations now have 2 paraeters Q(s,t) P (t) t=0.75 P 4 (t) t t=0.25 s (C) Dou Bowan, Virinia Tech,

29 (C) Dou Bowan, Virinia Tech, Matrix representation A sinle curve was expressed Q(t) = TM Now, the eoetric inforation varies Q(s,t) = SM " % & = ) ( ) ( ) ( ) ( t t t t each i (t) is itself a cubic curve

30 (C) Dou Bowan, Virinia Tech, Matrix representation (cont.) Since each i is a cubic curve, it can be written: i (t) = TM i " % & = i i i i i

31 Matrix representation (cont.) Substitutin, we obtain: Q(s,t) = SM = SMTM & % " & TM TM TM % TM (C) Dou Bowan, Virinia Tech, " This forulation does not work in ters of atrix diensions...

32 Matrix representation (cont.) So, use the transpose rule: i (t) = i T M T T T Q(s,t) = S M & & % M T T T = S M M T T T % 2 4 (C) Dou Bowan, Virinia Tech, T T 2 T 3 T 4 " "

33 Matrix representation (cont.) Finally, reeber that the lare eoetry atrix has 3 coponents (x, y, z) for each ij, so that we et three paraetric equations: x(s,t) = S M x M T T T y(s,t) = S M y M T T T z(s,t) = S M z M T T T (C) Dou Bowan, Virinia Tech,

34 Herite surfaces extension of Herite curves to paraetric bicubic surfaces four eleents of the eoetry atrix are now P (t), P 4 (t), R (t), R 4 (t) can be thouht of as interpolatin the curves Q(s,0) and Q(s,) or Q(0,t) and Q(,t) (C) Dou Bowan, Virinia Tech,

35 Herite surface atrices x(s,t) = S M Hx M T T T y(s,t) = S M Hy M T T T z(s,t) = S M Hz M T T T P x (t) = TM x x = " & & & & & % H x & = % (C) Dou Bowan, Virinia Tech, "

36 Renderin surfaces Can also use iterative ethods in s and t rid representation: render curves of constant t and constant s use perspective and hidden lines to indicate surface shape (C) Dou Bowan, Virinia Tech,

37 Renderin surfaces Solid representation: subdivide surface into flat sections render quadrilateral for each section shape indicated by shadin (need noral) (C) Dou Bowan, Virinia Tech,